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On a new class of refined discrete Hardy-type inequalities. (English) Zbl 1195.26039
Authors’ abstract: We state, prove and discuss a new refined general weighted discrete Hardy-type inequality with a non-negative kernel, related to an arbitrary non-negative convex (or positive concave) function on a real interval and to a positive real parameter. As its consequences, obtained by rewriting it for various suitably chosen parameters, kernels, weights and convex (or concave) functions, we derive new weighted and unweighted generalizations and refinements of some well-known inequalities such as Carleman’s inequality and the so-called Godunova’s inequality. Finally, by employing exponential and logarithmic convexity, as special cases of the usual convexity, we obtain some further refinements of the inequalities mentioned above.
MSC:
26D15Inequalities for sums, series and integrals of real functions
26B25Convexity and generalizations (several real variables)